Input/output systems

Module usage

An input/output system is defined as a dynamical system that has a system state as well as inputs and outputs (either inputs or states can be empty). The dynamics of the system can be in continuous or discrete time. To simulate an input/output system, use the input_output_response() function:

t, y = input_output_response(io_sys, T, U, X0, params)

An input/output system can be linearized around an equilibrium point to obtain a StateSpace linear system. Use the find_eqpt() function to obtain an equilibrium point and the linearize() function to linearize about that equilibrium point:

xeq, ueq = find_eqpt(io_sys, X0, U0)
ss_sys = linearize(io_sys, xeq, ueq)

Input/output systems can be created from state space LTI systems by using the LinearIOSystem class`:

io_sys = LinearIOSystem(ss_sys)

Nonlinear input/output systems can be created using the NonlinearIOSystem class, which requires the definition of an update function (for the right hand side of the differential or different equation) and and output function (computes the outputs from the state):

io_sys = NonlinearIOSystem(updfcn, outfcn, inputs=M, outputs=P, states=N)

More complex input/output systems can be constructed by using the interconnect() function, which allows a collection of input/output subsystems to be combined with internal connections between the subsystems and a set of overall system inputs and outputs that link to the subsystems:

steering = ct.interconnect(
    [plant, controller], name='system',
    connections=[['controller.e', '-plant.y']],
    inplist=['controller.e'], inputs='r',
    outlist=['plant.y'], outputs='y')

Interconnected systems can also be created using block diagram manipulations such as the series(), parallel(), and feedback() functions. The InputOutputSystem class also supports various algebraic operations such as * (series interconnection) and + (parallel interconnection).

Example

To illustrate the use of the input/output systems module, we create a model for a predator/prey system, following the notation and parameter values in FBS2e.

We begin by defining the dynamics of the system

import control
import numpy as np
import matplotlib.pyplot as plt

def predprey_rhs(t, x, u, params):
    # Parameter setup
    a = params.get('a', 3.2)
    b = params.get('b', 0.6)
    c = params.get('c', 50.)
    d = params.get('d', 0.56)
    k = params.get('k', 125)
    r = params.get('r', 1.6)

    # Map the states into local variable names
    H = x[0]
    L = x[1]

    # Compute the control action (only allow addition of food)
    u_0 = u if u > 0 else 0

    # Compute the discrete updates
    dH = (r + u_0) * H * (1 - H/k) - (a * H * L)/(c + H)
    dL = b * (a * H *  L)/(c + H) - d * L

    return [dH, dL]

We now create an input/output system using these dynamics:

io_predprey = control.NonlinearIOSystem(
    predprey_rhs, None, inputs=('u'), outputs=('H', 'L'),
    states=('H', 'L'), name='predprey')

Note that since we have not specified an output function, the entire state will be used as the output of the system.

The io_predprey system can now be simulated to obtain the open loop dynamics of the system:

X0 = [25, 20]                 # Initial H, L
T = np.linspace(0, 70, 500)   # Simulation 70 years of time

# Simulate the system
t, y = control.input_output_response(io_predprey, T, 0, X0)

# Plot the response
plt.figure(1)
plt.plot(t, y[0])
plt.plot(t, y[1])
plt.legend(['Hare', 'Lynx'])
plt.show(block=False)

We can also create a feedback controller to stabilize a desired population of the system. We begin by finding the (unstable) equilibrium point for the system and computing the linearization about that point.

eqpt = control.find_eqpt(io_predprey, X0, 0)
xeq = eqpt[0]                         # choose the nonzero equilibrium point
lin_predprey = control.linearize(io_predprey, xeq, 0)

We next compute a controller that stabilizes the equilibrium point using eigenvalue placement and computing the feedforward gain using the number of lynxes as the desired output (following FBS2e, Example 7.5):

K = control.place(lin_predprey.A, lin_predprey.B, [-0.1, -0.2])
A, B = lin_predprey.A, lin_predprey.B
C = np.array([[0, 1]])                # regulated output = number of lynxes
kf = -1/(C @ np.linalg.inv(A - B @ K) @ B)

To construct the control law, we build a simple input/output system that applies a corrective input based on deviations from the equilibrium point. This system has no dynamics, since it is a static (affine) map, and can constructed using the ~control.ios.NonlinearIOSystem class:

io_controller = control.NonlinearIOSystem(
  None,
  lambda t, x, u, params: -K @ (u[1:] - xeq) + kf * (u[0] - xeq[1]),
  inputs=('Ld', 'u1', 'u2'), outputs=1, name='control')

The input to the controller is u, consisting of the vector of hare and lynx populations followed by the desired lynx population.

To connect the controller to the predatory-prey model, we create an InterconnectedSystem using the interconnect() function:

io_closed = control.interconnect(
  [io_predprey, io_controller],       # systems
  connections=[
    ['predprey.u', 'control.y[0]'],
    ['control.u1',  'predprey.H'],
    ['control.u2',  'predprey.L']
  ],
  inplist=['control.Ld'],
  outlist=['predprey.H', 'predprey.L', 'control.y[0]']
)

Finally, we simulate the closed loop system:

# Simulate the system
t, y = control.input_output_response(io_closed, T, 30, [15, 20])

# Plot the response
plt.figure(2)
plt.subplot(2, 1, 1)
plt.plot(t, y[0])
plt.plot(t, y[1])
plt.legend(['Hare', 'Lynx'])
plt.subplot(2, 1, 2)
plt.plot(t, y[2])
plt.legend(['input'])
plt.show(block=False)

Additional features

The I/O systems module has a number of other features that can be used to simplify the creation of interconnected input/output systems.

Summing junction

The summing_junction() function can be used to create an input/output system that takes the sum of an arbitrary number of inputs. For ezample, to create an input/output system that takes the sum of three inputs, use the command

sumblk = ct.summing_junction(3)

By default, the name of the inputs will be of the form u[i] and the output will be y. This can be changed by giving an explicit list of names:

sumblk = ct.summing_junction(inputs=['a', 'b', 'c'], output='d')

A more typical usage would be to define an input/output system that compares a reference signal to the output of the process and computes the error:

sumblk = ct.summing_junction(inputs=['r', '-y'], output='e')

Note the use of the minus sign as a means of setting the sign of the input ‘y’ to be negative instead of positive.

It is also possible to define “vector” summing blocks that take multi-dimensional inputs and produce a multi-dimensional output. For example, the command

sumblk = ct.summing_junction(inputs=['r', '-y'], output='e', dimension=2)

will produce an input/output block that implements e[0] = r[0] - y[0] and e[1] = r[1] - y[1].

Automatic connections using signal names

The interconnect() function allows the interconnection of multiple systems by using signal names of the form sys.signal. In many situations, it can be cumbersome to explicitly connect all of the appropriate inputs and outputs. As an alternative, if the connections keyword is omitted, the interconnect() function will connect all signals of the same name to each other. This can allow for simplified methods of interconnecting systems, especially when combined with the summing_junction() function. For example, the following code will create a unity gain, negative feedback system:

P = control.tf2io(control.tf(1, [1, 0]), inputs='u', outputs='y')
C = control.tf2io(control.tf(10, [1, 1]), inputs='e', outputs='u')
sumblk = control.summing_junction(inputs=['r', '-y'], output='e')
T = control.interconnect([P, C, sumblk], inplist='r', outlist='y')

If a signal name appears in multiple outputs then that signal will be summed when it is interconnected. Similarly, if a signal name appears in multiple inputs then all systems using that signal name will receive the same input. The interconnect() function will generate an error if an signal listed in inplist or outlist (corresponding to the inputs and outputs of the interconnected system) is not found, but inputs and outputs of individual systems that are not connected to other systems are left unconnected (so be careful!).

Module classes and functions

Input/output system classes

InputOutputSystem([inputs, outputs, states, …])

A class for representing input/output systems.

InterconnectedSystem(syslist[, connections, …])

Interconnection of a set of input/output systems.

LinearICSystem(io_sys[, ss_sys])

Interconnection of a set of linear input/output systems.

LinearIOSystem(linsys[, inputs, outputs, …])

Input/output representation of a linear (state space) system.

NonlinearIOSystem(updfcn[, outfcn, inputs, …])

Nonlinear I/O system.

Input/output system functions

find_eqpt(sys, x0[, u0, y0, t, params, iu, …])

Find the equilibrium point for an input/output system.

linearize(sys, xeq[, ueq, t, params])

Linearize an input/output system at a given state and input.

input_output_response(sys, T[, U, X0, …])

Compute the output response of a system to a given input.

interconnect(syslist[, connections, …])

Interconnect a set of input/output systems.

ss2io(*args, **kwargs)

Create an I/O system from a state space linear system.

summing_junction([inputs, output, …])

Create a summing junction as an input/output system.

tf2io(*args, **kwargs)

Convert a transfer function into an I/O system